# How do you determine whether the sequence a_n=n-n^2/(n+1) converges, if so how do you find the limit?

Jul 27, 2017

The given Seq. converges to $1.$

#### Explanation:

Observe that, ${a}_{n} = n - {n}^{2} / \left(n + 1\right) , \left(n \in \mathbb{N}\right)$

$= \frac{n \left(n + 1\right) - {n}^{2}}{n + 1} ,$

$\Rightarrow {a}_{n} = \frac{n}{n + 1} .$

We know that, as $n \to \infty , \frac{1}{n} \to 0. \ldots \ldots \ldots \ldots \left(\ast\right) .$

$\text{ Therefore, the Reqd. Sequencial Limit=} {\lim}_{n \to \infty} {a}_{n} ,$

$= {\lim}_{n \to \infty} \frac{n}{n + 1} ,$

$= {\lim}_{n \to \infty} \frac{\cancel{n}}{\cancel{n} \left(1 + \frac{1}{n}\right)} ,$

$= {\lim}_{n \to \infty} \frac{1}{1 + \frac{1}{n}} ,$

$= \frac{1}{1 + 0} , \ldots \ldots \ldots \ldots \ldots . \left[\because , \left(\ast\right) .\right]$

$\text{ The Reqd. Sequencial Limit=} 1.$

Because, the Seq. Lim. exists, the seq. converges to $1.$